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Properties

Label 3840.2.f.h
Level $3840$
Weight $2$
Character orbit 3840.f
Analytic conductor $30.663$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(769,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} - q^{9} + ( - \beta_{4} - \beta_{3}) q^{11} + ( - \beta_{5} - \beta_{2} - 2 \beta_1) q^{13} - \beta_{5} q^{15} + (\beta_{4} - \beta_{3} - 2 \beta_1) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 2) q^{19}+ \cdots + (\beta_{4} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 2 q^{15} + 8 q^{19} - 8 q^{21} + 2 q^{25} - 8 q^{31} - 24 q^{35} + 12 q^{39} + 12 q^{41} - 22 q^{49} + 8 q^{51} - 32 q^{55} - 32 q^{59} - 24 q^{61} + 4 q^{65} + 16 q^{69} - 16 q^{71} + 8 q^{75} - 40 q^{79} + 6 q^{81} + 24 q^{85} - 20 q^{89} + 32 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.403032 + 0.403032i
−0.854638 − 0.854638i
1.45161 + 1.45161i
0.403032 − 0.403032i
−0.854638 + 0.854638i
1.45161 − 1.45161i
0 − 1.00000i 0 −1.67513 − 1.48119i 0 − 4.96239i 0 −1.00000 0
769.2 0 − 1.00000i 0 −0.539189 + 2.17009i 0 2.34017i 0 −1.00000 0
769.3 0 − 1.00000i 0 2.21432 + 0.311108i 0 − 1.37778i 0 −1.00000 0
769.4 0 1.00000i 0 −1.67513 + 1.48119i 0 4.96239i 0 −1.00000 0
769.5 0 1.00000i 0 −0.539189 − 2.17009i 0 − 2.34017i 0 −1.00000 0
769.6 0 1.00000i 0 2.21432 − 0.311108i 0 1.37778i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.f.h 6
4.b odd 2 1 3840.2.f.e 6
5.b even 2 1 inner 3840.2.f.h 6
8.b even 2 1 3840.2.f.g 6
8.d odd 2 1 3840.2.f.f 6
16.e even 4 1 1920.2.d.c ✓ 6
16.e even 4 1 1920.2.d.f yes 6
16.f odd 4 1 1920.2.d.d yes 6
16.f odd 4 1 1920.2.d.e yes 6
20.d odd 2 1 3840.2.f.e 6
40.e odd 2 1 3840.2.f.f 6
40.f even 2 1 3840.2.f.g 6
80.k odd 4 1 1920.2.d.d yes 6
80.k odd 4 1 1920.2.d.e yes 6
80.q even 4 1 1920.2.d.c ✓ 6
80.q even 4 1 1920.2.d.f yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.d.c ✓ 6 16.e even 4 1
1920.2.d.c ✓ 6 80.q even 4 1
1920.2.d.d yes 6 16.f odd 4 1
1920.2.d.d yes 6 80.k odd 4 1
1920.2.d.e yes 6 16.f odd 4 1
1920.2.d.e yes 6 80.k odd 4 1
1920.2.d.f yes 6 16.e even 4 1
1920.2.d.f yes 6 80.q even 4 1
3840.2.f.e 6 4.b odd 2 1
3840.2.f.e 6 20.d odd 2 1
3840.2.f.f 6 8.d odd 2 1
3840.2.f.f 6 40.e odd 2 1
3840.2.f.g 6 8.b even 2 1
3840.2.f.g 6 40.f even 2 1
3840.2.f.h 6 1.a even 1 1 trivial
3840.2.f.h 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3840, [\chi])\):

\( T_{7}^{6} + 32T_{7}^{4} + 192T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} - 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{29}^{3} - 16T_{29} - 16 \) Copy content Toggle raw display
\( T_{31}^{3} + 4T_{31}^{2} - 32T_{31} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 44 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$17$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 64 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{3} - 16 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} - 32 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 172 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$41$ \( (T - 2)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 176 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{6} + 96 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$53$ \( T^{6} + 156 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{3} + 16 T^{2} - 592)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 304 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 192 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$79$ \( (T^{3} + 20 T^{2} - 1184)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 304 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} + \cdots - 712)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 192 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
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